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arXiv · 2509.20836

Locally integrable cross sections and their intersection covolume

Nachi Avraham-Re'em·Michael Björklund·Rickard Cullman
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Abstract

We study systematically cross sections of probability preserving actions of unimodular groups and their associated transverse measures, and introduce the invariant \emph{intersection covolume} to quantify their periodicity. Our main theorem, derived from a higher order version of Kac's lemma, shows that the intersection covolume is bounded below by the intensity, with equality precisely when the action is induced by a lattice (in the sense of Mackey). We further prove that the natural cross sections of cut--and--project actions have finite intersection covolume.

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Nachi Avraham-Re'em, Michael Björklund, Rickard Cullman. 2025-09-25. Locally integrable cross sections and their intersection covolume. https://arxiv.org/abs/2509.20836

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